tailieunhanh - Genetic Genealogical Models in Rare Event Analy- sis

The estimation of these quantites arises in many research areas such as in physics and engineering problems. In network analysis such as in advanced telecommunication systems studies X traditionally represents the length of service centers in an open/closed queueing network processing jobs. In this context these two quantities () represent repectively the probability of bu er-over ows and the distribution of the queueing process in this over ow regime. Several numerical methods have been proposed in the literature to estimate the entrance probability into a rare set. We refer the reader to the excellent paper Glasserman et al. (1999) which contains a precise review on these methods as well as a detailed. | Alea 1 181-203 2006 Genetic Genealogical Models in Rare Event Analysis Frédéric Cérou Pierre Del Moral Francois LeGland and Pascal Lezaud IRISA INRIA Campus de Beaulieu 35042 RENNES Cédex France E-mail address Laboratoire . Dieudonné Université Nice Sophia-Antipolis Parc Valrose 06108 NICE Céedex 2 France E-mail address delmoral@ IRISA INRIA Campus de Beaulieu 35042 RENNES Céedex France E-mail address legland@ Centre d Etudes de la Navigation Aérienne 7 avenue Edouard Belin 31055 TOULOUSE Céedex 4 France E-mail address lezaud@ Abstract. We present in this article a genetic type interacting particle systems algorithm and a genealogical model for estimating a class of rare events arising in physics and network analysis. We represent the distribution of a Markov process hitting a rare target in terms of a Feynman-Kac model in path space. We show how these branching particle models described in previous works can be used to estimate the probability of the corresponding rare events as well as the distribution of the process in this regime. 1. Introduction Let X Xt t 0g be a continuous-time strong Markov process taking values in some Polish state space S. For a given target Borel set B c S we define the hitting time Tb inf t 0 Xt 2 Bg as the first time when the process X hits B. Let us assume that X has almost surely right continuous left limited trajectories RCLL and that B is closed. Then we Received by the editors August 31 2005 accepted April 5 2006. 2000 Mathematics Subject Classification. 65C35. Key words and phrases. interacting particle systems rare events Feynman-Kac models genetic algorithms genealogical trees. Second version in which misprints have been corrected. 181 182 Frédéric Cérou et al. have that XTb 2 B. In many applications the set B is the super level set of a scalar measurable function Ộ defined on S . B x 2 S x Xbg In this case we will assume that Ộ is upper semi-continuous which ensures .

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